Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on RN

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$ by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most $p$-times integrability where $p$ is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ with respect to the initial values belonging to $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$. As an application of the result on the continuity, the existence of a pullback attractor in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ is proved for arbitrary $N\ge 1$ and $p>2$.     

Existence of ground state solutions of Nehari–Pohozaev type for fractional Schrödinger–Poisson systems with a general potential

In this paper, we consider the following fractional Schrödinger–Poissonproblem

where $0<s\le t<1$ and $2s+2t>3$, the potential $V\left(x\right)$ is weakly differentiable and $f\in C(\mathbb{R},\mathbb{R})$. By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on $V$ and $f$. The results here extend the existing study.     

Multiplicity of positive solutions for the Kirchhoff-type equations with critical exponent in RN

In this work we study the existence and multiplicity of solutions to the following Kirchhoff-type problem with critical nonlinearity in ${\mathbb{R}}^N$

where $N\ge 2p$, $\mu ,\lambda ,a,b>0$ and the nonlinearity $f(x,u)$ satisfies certain subcritical growth conditions. By using topological and variational methods, infinitely many positive solutions are obtained.     

Existence of ground states for a Kirchhoff type problem without 4-superlinear condition

We consider the existence of ground state solutions for the Kirchhoff type problem

where $a,b>0$, $N=1,2,3$ and $2<p<2^1$. Here we are interested in the case that $2<p\le 4$ since the existence of ground state for $4<p\le 2^1$ is easily obtained by a standard variational argument. Our method is based on a Poho$\stackrel{?}{z}$aev type identity.     

Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega ?{\mathbb{R}}^2$, where the parameters $\chi ,\mu >0$ and $r\in \mathbb{R}$. Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.